Bayesian analysis of Organic Orthogonal Phase Seperation Data Part 2

Oliver M. Crook

1 Introduction

Here, we attempt to build a principled Bayesian model for some OOPS data. We illustrate the main ideas of a Bayesian workflow and refer to the accompanying manuscript for relevent details. The first code chunk loads the packages we are interested in.

library(brms)

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library(pheatmap)
library(RColorBrewer)
library(MSnbase)

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library(ggplot2)

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library(ggfortify)

Now, we load the data we are interested, which is from Queiroz et al. 2019. The data have already been summarised to protein level and column median normalised. The design used 2 seperate TMT 10-plexes to measure protein abundance and oops sample. This allows us to model changes in RNA binding that are indepedent of concurrent changes in total protien abundance. Changes in oops-enriched protein abundance samples that do not correlated with total protein abundance chances are indicative of specific RBPs binding RNA differentially in those conditions. The experimental design uses thymidine-nocodazole arrest and then measures protein abundance, as well as oops enriched protein abundance at \(0,6\) and \(23\) hours. Triplicate measurements are made, except for at 6 hours where 4 replicates where taken. For further details see Queiroz et al. 2019

# We assume the data is in the working directory
oopsdata <- readMSnSet2(file = "oopsdata.csv", ecol = 2:21, fnames = 1)

We test using a single protein for simplicity

set.seed(1)
data <- data.frame(exprs(oopsdata)[sample.int(nrow(oopsdata), size = 1),])
colnames(data) <- "abundance"
data$type <- as.factor(rep(c("total", "oops"), each = 10))
data$time <- as.factor(rep(rep(c("0", "6", "23"), times = c(3,4,3)), 2))
data$replicate <- as.factor(sapply(strsplit(rownames(data), "_"), function(x)x[2]))
data$tag <- as.factor(rep(TMT10@reporterNames, times = 2))

2 More complex modelling

We previously left aside some of the modelling aspects to explore the full workflow. Let us explore building and evaluating a more bespoke model, than captures more of the variations in the data. The first point we raised was that the variance was different for the oops samples, total samples and at different times. In the following we allow \(\sigma\) to vary with sample type and time. Note we are now on the log scale and so move to a normal prior for \(\log \sigma\).

pr <- c(prior(normal(0,1), class = "b"),
  prior(normal(0, 3), dpar = "sigma"))


fit_prot1_post2 <- brm(bf(abundance ~ 0 + type + time + (type:time),
                     sigma ~ 0 + type + time),
                 data = data,
                 family = gaussian(link = "identity"),
                 sample_prior = "no",
                 prior = pr)

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plot(fit_prot1_post2, ask = FALSE)

summary(fit_prot1_post2)

##  Family: gaussian 
##   Links: mu = identity; sigma = log 
## Formula: abundance ~ 0 + type + time + (type:time) 
##          sigma ~ 0 + type + time
##    Data: data (Number of observations: 20) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## typeoops             0.08      0.05    -0.02     0.18 1.00     1464     1332
## typetotal            0.30      0.04     0.21     0.38 1.00     1950     1227
## time23              -0.28      0.13    -0.56    -0.02 1.01     1467     1476
## time6                0.07      0.06    -0.04     0.20 1.00     1421     1415
## typetotal:time23     0.13      0.17    -0.21     0.51 1.00     1449     1235
## typetotal:time6     -0.24      0.08    -0.41    -0.07 1.00     1383     1514
## sigma_typeoops      -2.69      0.52    -3.55    -1.51 1.00      977     1107
## sigma_typetotal     -2.84      0.46    -3.56    -1.74 1.00      911     1003
## sigma_time23         0.97      0.58    -0.22     2.06 1.00     1116     1241
## sigma_time6          0.04      0.55    -1.13     1.05 1.00     1026      966
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

We notice that the estimated errors for some of the effects have become larger in this more complex. The more complex model has become harder to estimate and therefore our inferences are likely to be unreliable. The first remedy is to provide stronger prior information. We already saw in previous modelling that our priors we quite diffuse. Here, we place even stronger prior information on \(\log \sigma\).

pr <- c(prior(normal(0,1), class = "b"),
  prior(normal(0,1), dpar = "sigma"))


fit_prot1_post2a <- brm(bf(abundance ~ 0 + type + time + (type:time),
                     sigma ~ 0 + type + time),
                 data = data,
                 family = gaussian(link = "identity"),
                 sample_prior = "no",
                 prior = pr,
                 control = list(adapt_delta = 0.99))

## Compiling Stan program...

## Start sampling

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plot(fit_prot1_post2a, ask = FALSE)

summary(fit_prot1_post2a)

##  Family: gaussian 
##   Links: mu = identity; sigma = log 
## Formula: abundance ~ 0 + type + time + (type:time) 
##          sigma ~ 0 + type + time
##    Data: data (Number of observations: 20) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## typeoops             0.08      0.11    -0.17     0.31 1.01      963     1004
## typetotal            0.31      0.10     0.11     0.52 1.00     1362     1056
## time23              -0.27      0.16    -0.59     0.04 1.00     1157     1198
## time6                0.08      0.12    -0.17     0.33 1.00      946      970
## typetotal:time23     0.12      0.21    -0.30     0.54 1.00     1096     1305
## typetotal:time6     -0.25      0.16    -0.57     0.06 1.00      868      932
## sigma_typeoops      -1.83      0.51    -2.78    -0.80 1.01      888     1880
## sigma_typetotal     -2.03      0.48    -2.92    -1.02 1.00      844     1513
## sigma_time23         0.11      0.52    -0.90     1.12 1.00     1059     1808
## sigma_time6         -0.73      0.50    -1.71     0.29 1.00     1007     2061
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

# compute kernel density estimate of data and simulated data and plot
pp_check(object = fit_prot1_post2a, "dens_overlay") 

## Using 10 posterior samples for ppc type 'dens_overlay' by default.

# plot data in a histogram to see distributions
pp_check(object = fit_prot1_post2a, "hist")

## Using 10 posterior samples for ppc type 'hist' by default.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# plot data in a boxplot to see distribtuions
pp_check(object = fit_prot1_post2a, "boxplot")

## Using 10 posterior samples for ppc type 'boxplot' by default.

## notch went outside hinges. Try setting notch=FALSE.

# produce summary statistics
pp_check(object = fit_prot1_post2a, "stat_2d")

## Using all posterior samples for ppc type 'stat_2d' by default.

# produce intervals
pp_check(object = fit_prot1_post2a, "intervals")

## Using all posterior samples for ppc type 'intervals' by default.

# predctive errors
pp_check(object = fit_prot1_post2a, "error_hist")

## Using 10 posterior samples for ppc type 'error_hist' by default.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

We still see that the estimated errors are quite large for the data with excessive mass in the extremes of the distribution. We need to think more carefully about the model we are trying to fit. The normal prior does not allow much variation in the scale of \(\sigma\) and given that we are now allowing the standard deviations to vary between samples our confidence is the scale is more uncertain. Our current modelling approach has not captured this and so we need to use a prior that reflects this uncertainty. We can do this using a heavy tailed prior on \(\sigma\). We opt for a student-t prior as below, note that sigma is now

pr <- c(prior(normal(0, 1), class = "b"),
  prior(student_t(3, 0, 1), dpar = "sigma"))


fit_prot1_post2b <- brm(bf(abundance ~ 0 + type + time + (type:time),
                     sigma ~ 0 + type + time),
                 data = data,
                 family = gaussian(link = "identity"),
                 sample_prior = "no",
                 prior = pr,
                 control = list(adapt_delta = 0.99),
                 save_pars = save_pars(all = TRUE))

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plot(fit_prot1_post2b, ask = FALSE)

summary(fit_prot1_post2b)

##  Family: gaussian 
##   Links: mu = identity; sigma = log 
## Formula: abundance ~ 0 + type + time + (type:time) 
##          sigma ~ 0 + type + time
##    Data: data (Number of observations: 20) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## typeoops             0.08      0.09    -0.10     0.26 1.01     1027      963
## typetotal            0.30      0.07     0.17     0.44 1.00     1521     1261
## time23              -0.27      0.15    -0.56     0.04 1.00     1356     1396
## time6                0.08      0.10    -0.11     0.27 1.01     1066      980
## typetotal:time23     0.11      0.19    -0.27     0.47 1.00     1376     1308
## typetotal:time6     -0.25      0.12    -0.50    -0.00 1.01     1036     1010
## sigma_typeoops      -2.24      0.56    -3.19    -1.04 1.01      703     1036
## sigma_typetotal     -2.44      0.54    -3.31    -1.27 1.01      708     1023
## sigma_time23         0.48      0.56    -0.65     1.52 1.01      934     1209
## sigma_time6         -0.36      0.54    -1.52     0.63 1.01      821     1491
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

We can again perform posterior predictive checks to see if our model has good predictive qualities. We can visually see that the model is capturing the behaviour of the data extremely well.

# compute kernel density estimate of data and simulated data and plot
pp_check(object = fit_prot1_post2b, "dens_overlay") 

## Using 10 posterior samples for ppc type 'dens_overlay' by default.

# plot data in a histogram to see distributions
pp_check(object = fit_prot1_post2b, "hist")

## Using 10 posterior samples for ppc type 'hist' by default.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# plot data in a boxplot to see distribtuions
pp_check(object = fit_prot1_post2b, "boxplot")

## Using 10 posterior samples for ppc type 'boxplot' by default.

## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.

# produce summary statistics
pp_check(object = fit_prot1_post2b, "stat_2d")

## Using all posterior samples for ppc type 'stat_2d' by default.

# produce intervals
pp_check(object = fit_prot1_post2b, "intervals")

## Using all posterior samples for ppc type 'intervals' by default.

# predctive errors
pp_check(object = fit_prot1_post2b, "error_hist")

## Using 10 posterior samples for ppc type 'error_hist' by default.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Another aspect of the model that we haven’t consider is the replicate structure. The replicate structure is a grouping that can be included as a group-level effect. We can again specify the prior on the standard deviation of this random effect.

pr <- c(prior(normal(0, 1), class = "b"),
  prior(student_t(3, 0, 1), dpar = "sigma"),
  prior(student_t(3, 0, 0.1), class = "sd"))


fit_prot1_post2c <- brm(bf(abundance ~ 0 + type + time + (type:time) + (1|replicate),
                     sigma ~ 0 + type + time),
                 data = data,
                 family = gaussian(link = "identity"),
                 sample_prior = "no",
                 prior = pr,
                 control = list(adapt_delta = 0.99),
                 save_pars = save_pars(all = TRUE))

## Compiling Stan program...

## Start sampling

## 
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## Warning: There were 4 divergent transitions after warmup. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## to find out why this is a problem and how to eliminate them.

## Warning: Examine the pairs() plot to diagnose sampling problems

plot(fit_prot1_post2c, ask = FALSE)

summary(fit_prot1_post2c)

## Warning: There were 4 divergent transitions after warmup. Increasing adapt_delta
## above 0.99 may help. See http://mc-stan.org/misc/warnings.html#divergent-
## transitions-after-warmup

##  Family: gaussian 
##   Links: mu = identity; sigma = log 
## Formula: abundance ~ 0 + type + time + (type:time) + (1 | replicate) 
##          sigma ~ 0 + type + time
##    Data: data (Number of observations: 20) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Group-Level Effects: 
## ~replicate (Number of levels: 4) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.06      0.05     0.00     0.17 1.00      833     1006
## 
## Population-Level Effects: 
##                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## typeoops             0.10      0.07    -0.06     0.24 1.00     1433     1058
## typetotal            0.31      0.08     0.16     0.47 1.00     1288     1176
## time23              -0.28      0.13    -0.52    -0.01 1.01     1877     1191
## time6                0.05      0.07    -0.07     0.19 1.00     2037     1442
## typetotal:time23     0.13      0.19    -0.25     0.50 1.00     1659     1616
## typetotal:time6     -0.23      0.09    -0.42    -0.04 1.00     1405     1299
## sigma_typeoops      -2.69      0.68    -4.03    -1.31 1.00      830     1327
## sigma_typetotal     -2.56      0.59    -3.61    -1.30 1.00     1252     1626
## sigma_time23         0.79      0.62    -0.39     2.00 1.00     1096     1600
## sigma_time6         -0.50      0.58    -1.71     0.56 1.00     1365     1749
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

There are a few divergent transitions suggesting that some aspects of the model might be computational unfaithful. Given that there are only few we do not worry about this here. We continue to perform posterior predictive checks.

# compute kernel density estimate of data and simulated data and plot
pp_check(object = fit_prot1_post2c, "dens_overlay") 

## Using 10 posterior samples for ppc type 'dens_overlay' by default.

# plot data in a histogram to see distributions
pp_check(object = fit_prot1_post2c, "hist")

## Using 10 posterior samples for ppc type 'hist' by default.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# plot data in a boxplot to see distribtuions
pp_check(object = fit_prot1_post2c, "boxplot")

## Using 10 posterior samples for ppc type 'boxplot' by default.

## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.

# produce summary statistics
pp_check(object = fit_prot1_post2c, "stat_2d")

## Using all posterior samples for ppc type 'stat_2d' by default.

# produce intervals
pp_check(object = fit_prot1_post2c, "intervals")

## Using all posterior samples for ppc type 'intervals' by default.

# predctive errors
pp_check(object = fit_prot1_post2c, "error_hist")

## Using 10 posterior samples for ppc type 'error_hist' by default.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

The posterior predictive checks look good. However, is this really a better model than the previous models. Using probability, we can actually quantify the extent to which this model is preferred.

post_prob(fit_prot1_post2b,
          fit_prot1_post2c,
          prior_prob = c(0.5,0.5),
          model_names = c("no random effect", "random effect"))

## Iteration: 1
## Iteration: 2
## Iteration: 3
## Iteration: 4
## Iteration: 5
## Iteration: 6
## Iteration: 1
## Iteration: 2
## Iteration: 3
## Iteration: 4
## Iteration: 5
## Iteration: 6

## no random effect    random effect 
##        0.4118754        0.5881246

The computation here suggest that the model with random effects is preferred but the probability is only about \(0.6\). We can also use predictive cross-validation checks to see if this is a better model. We use leave-one-out (loo) cross-validation with log predictive density as the utility function to evaluate the models.

loo_compare(loo(fit_prot1_post2b, moment_match = TRUE, reloo = TRUE),
            loo(fit_prot1_post2c, moment_match = TRUE, reloo = TRUE))

## Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.

## No problematic observations found. Returning the original 'loo' object.

## Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.

## No problematic observations found. Returning the original 'loo' object.

##                  elpd_diff se_diff
## fit_prot1_post2c  0.0       0.0   
## fit_prot1_post2b -1.9       1.4

Here we can see that according to loo-cv that the model with random effects is preferred. We can now add a random effect according to the TMT tag that was used.

pr <- c(prior(normal(0, 1), class = "b"),
  prior(student_t(3, 0, 1), dpar = "sigma"),
  prior(student_t(3, 0, 0.1), class = "sd"))


fit_prot1_post2d <- brm(bf(abundance ~ 0 + type + time + (type:time) + 
                             (1|replicate) + (1|tag),
                     sigma ~ 0 + type + time),
                 data = data,
                 family = gaussian(link = "identity"),
                 sample_prior = "no",
                 prior = pr,
                 control = list(adapt_delta = 0.99),
                 save_pars = save_pars(all = TRUE))

## Compiling Stan program...

## Start sampling

## 
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## Warning: There were 8 divergent transitions after warmup. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## to find out why this is a problem and how to eliminate them.

## Warning: Examine the pairs() plot to diagnose sampling problems

plot(fit_prot1_post2d, ask = FALSE)

summary(fit_prot1_post2d)

## Warning: There were 8 divergent transitions after warmup. Increasing adapt_delta
## above 0.99 may help. See http://mc-stan.org/misc/warnings.html#divergent-
## transitions-after-warmup

##  Family: gaussian 
##   Links: mu = identity; sigma = log 
## Formula: abundance ~ 0 + type + time + (type:time) + (1 | replicate) + (1 | tag) 
##          sigma ~ 0 + type + time
##    Data: data (Number of observations: 20) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Group-Level Effects: 
## ~replicate (Number of levels: 4) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.05      0.04     0.00     0.15 1.00     1400     1839
## 
## ~tag (Number of levels: 10) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.03      0.03     0.00     0.10 1.00     1338     1446
## 
## Population-Level Effects: 
##                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## typeoops             0.09      0.08    -0.08     0.24 1.00     1729     1524
## typetotal            0.31      0.08     0.15     0.47 1.00     2146     1524
## time23              -0.27      0.14    -0.54     0.01 1.00     2264     1675
## time6                0.06      0.08    -0.09     0.24 1.00     1896     1462
## typetotal:time23     0.12      0.19    -0.27     0.50 1.00     2246     1982
## typetotal:time6     -0.24      0.10    -0.44    -0.03 1.00     1852     1392
## sigma_typeoops      -2.52      0.69    -3.90    -1.18 1.00     1156     1440
## sigma_typetotal     -2.50      0.63    -3.68    -1.20 1.00     1172     1656
## sigma_time23         0.67      0.61    -0.51     1.90 1.00     1574     2211
## sigma_time6         -0.62      0.60    -1.86     0.46 1.00     1777     2010
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Again, we can examine posterior predictive checks, posterior model probabilities and loo evaluation of the previous models.

# compute kernel density estimate of data and simulated data and plot
pp_check(object = fit_prot1_post2d, "dens_overlay") 

## Using 10 posterior samples for ppc type 'dens_overlay' by default.

# plot data in a histogram to see distributions
pp_check(object = fit_prot1_post2d, "hist")

## Using 10 posterior samples for ppc type 'hist' by default.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# plot data in a boxplot to see distribtuions
pp_check(object = fit_prot1_post2d, "boxplot")

## Using 10 posterior samples for ppc type 'boxplot' by default.

## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.
## notch went outside hinges. Try setting notch=FALSE.

# produce summary statistics
pp_check(object = fit_prot1_post2d, "stat_2d")

## Using all posterior samples for ppc type 'stat_2d' by default.

# produce intervals
pp_check(object = fit_prot1_post2d, "intervals")

## Using all posterior samples for ppc type 'intervals' by default.

# predctive errors
pp_check(object = fit_prot1_post2d, "error_hist")

## Using 10 posterior samples for ppc type 'error_hist' by default.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

post_prob(fit_prot1_post2b,
          fit_prot1_post2c,
          fit_prot1_post2d,
          prior_prob = c(1/3,1/3, 1/3),
          model_names = c("no random effect",
                          "random effect",
                          "two random effects"))

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##   no random effect      random effect two random effects 
##          0.3543369          0.4753413          0.1703218

loo_compare(loo(fit_prot1_post2b, moment_match = TRUE, reloo = TRUE),
            loo(fit_prot1_post2c, moment_match = TRUE, reloo = TRUE),
            loo(fit_prot1_post2d, moment_match = TRUE, reloo = TRUE))

## Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.

## No problematic observations found. Returning the original 'loo' object.

## Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.

## No problematic observations found. Returning the original 'loo' object.

## Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.

## No problematic observations found. Returning the original 'loo' object.

##                  elpd_diff se_diff
## fit_prot1_post2d  0.0       0.0   
## fit_prot1_post2c -0.4       1.0   
## fit_prot1_post2b -2.4       1.6

From the above analysis we can see that when using posterior model probabilities the model with random effect according to replicates is preferred with the model using random effects for replicates and tags the least preferred model. However, when using LOO-CV the preferred model is the model with random effects for tags and replicates followed shortly by the other model with random effects.

The question now is with which model do we progress. Ultimately,this will come down to what we want the model to do. However, it appears that there is not a clear case for the more complex random effects model and so favoring simplicity the simpler random effects model could be justified. If users were so inclined they could also average these models using the posterior model probabilities as weights.

We return to our original question of whether there is a change in proportion of the total protein bound to RNA. We examine the posterior distribution from the interactions of interest. We can plot the joint distribution of the two interactions

plot(hexbin::hexbin(posterior_samples(fit_prot1_post2c)[,5:6]),
     colramp = colorRampPalette(brewer.pal(11, "PRGn")),
     xlab = "total:time23h", ylab = "total:time6h")

It is clear that the effects of these interactions are in the opposite directions for each time. We can compute the probability that these interactions are < 0 and > 0 respectively. We can also ask about specific effect-sizes we are interested in. We note that we are slightly more confident that the interaction at 6h is less than \(-0.1\) in our original analysis likely due to our account for more variability in our model through the random effect.

# Probability greater than 0 interaction at 23h
sum(posterior_samples(fit_prot1_post2c)[,5] > 0)/length(posterior_samples(fit_prot1_post2c)[,5])

## [1] 0.79525

# Probability less than 0 interaction at 6h
sum(posterior_samples(fit_prot1_post2c)[,6]  < 0)/length(posterior_samples(fit_prot1_post2c)[,6])

## [1] 0.98375

# probability less than -0.1 interaction 6h
sum(posterior_samples(fit_prot1_post2c)[,6]  < -0.1)/length(posterior_samples(fit_prot1_post2c)[,6])

## [1] 0.938

As always, we could further extend this model. Ask more questions of these probability distributions, perform model averaging. We could continue to apply this model to other proteins.